The Stacks project

Exercise 111.32.5. Let $j : U \to X$ be an open immersion. Show that

1. Pullback $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X) \to \mathop{\mathit{Sh}}\nolimits (U)$ has a left adjoint $j_{!} : \mathop{\mathit{Sh}}\nolimits (U) \to \mathop{\mathit{Sh}}\nolimits (X)$ called extension by the empty set.

2. Characterize the stalks of $j_{!}({\mathcal G})$ for $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (U)$.

3. Pullback $j^{-1} : \textit{Ab}(X) \to \textit{Ab}(U)$ has a left adjoint $j_{!} : \textit{Ab}(U) \to \textit{Ab}(X)$ called extension by zero.

4. Characterize the stalks of $j_{!}({\mathcal G})$ for $\mathcal{G} \in \textit{Ab}(U)$.

Observe that extension by zero differs from extension by the empty set!